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  • Patricia Scheler

    January 23, 2022 at 8:55 pm

    My big take away is to provide opportunities for multiple representations of numbers. I already use that in our number talks. I’m laughing, however, at this example. I followed everything, but my 2nd graders would not. I’m trying to give them a solid foundation in math, but this example was certainly not fitting. Sorry.

    • Kyle Pearce

      January 24, 2022 at 6:56 am

      Hi Patricia!

      Can you elaborate on the example and what wasn’t a good fit? Would love to better understand?

  • Patricia Scheler

    February 3, 2022 at 7:03 pm

    The example started with the speaker’s daughter’s shirt (with hearts on it). The example extended to algebraic equations. I thought it was kind of cool, but I have kids who are still using their fingers to count back for subtraction. This is not relevant to me.

    • Kyle Pearce

      February 7, 2022 at 6:47 am

      Patricia – while extending to algebraic equations is not immediately relevant, the point is… we explore patterns with students from pre-k through grade 12 and it is important for each educator to understand the journey so you can nudge students further and further along that progression. The question becomes… what prompt / questions might you have for your grade 2s that you can ask to nudge them one step further?

  • Dawn Oliver

    February 22, 2022 at 1:42 pm

    It always amazes me how a simple pattern can be represented mathematically. I could definitely use this type of model to teach polynomials. I could use an example like this for a Notice and Wonder, so it will lead into a lesson on how to multiply polynomials or come up with factors for a polynomial. This model could also be used in Geometry to teach composite figures and areas.

    • This reply was modified 11 months, 2 weeks ago by  Dawn Oliver.
  • Stephanie Pritchett

    February 24, 2022 at 10:25 pm

    I love showing the connection between linear equations in tables, graphs, and equations. It is helpful to see how these different representations show the same patterns. Today I used the Pentamino activity and I think it was a great way to show different ways to represent the same equation. I had a lot of push back from students. Of course challenges in listening, following along and are all part of the challenge in a special education class, however I still was met with resistance to learn or try something new compared to guessing and checking. Students were curious in the guessing a finding the numbers that made the totals, but struggled engaging in other strategies.

    • Kyle Pearce

      February 25, 2022 at 6:33 am

      Glad to hear you gave it a shot! It can be really challenging to get students to push themselves out of their comfort zone… but by doing these things consistently, they build the mindset to keep going!

  • Zorica Lloyd

    February 26, 2022 at 2:40 am

    Multiple representations is one of my favorite tools. I think a big take away is having students explore the equivalence of the various representations. We may see. things differently, but in the end it’s all the same. The other important step, for me, is to go back and map the representation to the original problem it represents (ex. if the is an n^2 in your expression you will see can find an n x n square in your drawing).

    • This reply was modified 11 months, 2 weeks ago by  Zorica Lloyd.
  • Tarini Arte

    March 3, 2022 at 1:46 am

    I’ve always loved this type of problem that allows students multiple “right” answers or approaches. When I’ve taught high school functions in the past, I would start off the year with just printing a bunch of these for the whole class and they could draw all over them to figure out the next step or the algebraic equation (or the 57th step)…

    Kind of a combination of manipulatives (concrete) and multiple representations by having them draw all over the shapes with colored pencils/shade in and show the solution in whatever way they see it. Nice discussions too!

  • Kerri Brodie

    March 9, 2022 at 7:35 pm

    I really like looking at the Algebra this way. It allows students to access the math in many different ways that are all correct and it allows them to see a physical representation of squaring, mutliplying, adding, subtracting, etc. and connect them all together. I think this would be a great way to start a lesson on Polynomials and it would certainly spark curiosity. In addition, it’s easier to refer back to “remember when we did that problem with the hearts?” than “remember when I drew those squares on the board?” when you are trying to call on prior knowledge!

  • Gwyneth McIntosh

    March 10, 2022 at 6:17 pm

    We work on multiple representations and perspectives throughout our class. At times we come across student resistance to representing in different ways. They can be a bit rigid…I think as we continue to build a culture that is open to thinking and pushing ourselves to see multiple perspectives and ways of representing, we will see less resistance. Any suggestions to help support the resistant ones?

    • Kyle Pearce

      March 10, 2022 at 8:10 pm

      Resistance can come from a number of reasons. One of which is the lack of clarity around why we are asking them to show multiple representations. When we help students understand that being mathematically proficient involves representing their thinking in more ways than one, the resistance weakens.

      • Gwyneth McIntosh

        March 14, 2022 at 8:50 pm

        I agree…we talk about how learning and seeing different ways (and working through the struggle to see these different ways) can also help us solve problems in the future. The more tools/strategies we have at our finger tips, the more entry points we have to solve challenging problems. We’ll keep up the sell! 🙂

  • Jonathan Lind

    March 12, 2022 at 8:58 am

    Multiple representations is something that’s actually on the rubrics we use to assess in math class. We encourage visual, algebraic, numeric, and verbal representations in the process of problem solving, and especially in the act of communicating reasoning. Students aren’t always used to doing this, but when they get into it, we see some innovative, novel ways of representing problems, and of connecting those representations together. Being able to make these connections makes the students better communicators, reasoners, and problem solvers in general.

  • David McKnight

    March 12, 2022 at 11:55 pm

    I like to open the discussion on these sort of questions. Kids usually reveal a wide array of strategies within a given class. It also helps to give questions that don’t necessarily have one correct answer.

  • Stephen Levesque

    March 14, 2022 at 10:55 am

    I really like the idea of multiple representations to help students make connections. It can feel time-consuming, but the payoff in student understanding and sense-making is well worth it. As the teacher, I want to make sure that the representations and connections we discuss are appropriate for the students’ mathematical maturity.

  • Lizann “Lizzie” Herrera

    March 17, 2022 at 2:01 am

    This lesson blew my mind! When I was in the classroom as a third grade teacher, we began working with distributive property. This would be a great way to create a deep understanding of this! And the fact that this lesson can be extended to be a grade 9 or above lesson, it’s amazing! I think is important to have a clear lesson goal so that you don’t go down a rabbit hole with this depending on how far you need the lesson to go. Thanks for sharing this!

  • Terry Hill

    April 2, 2022 at 2:19 pm

    There is clearly a need to use multiple representations as a way to get
    students to grasp the lesson being taught. Interestingly, I used this in
    a different way. Since day 1 had 2 hearts, day 2 had 7 hearts, and day 3
    had 14 hearts, on my day 4 there would have been 23 hearts, but I did
    it by increasing by 2 each day, after day one, how many hearts were added. So day one would have 2, day two 2 + 5, day three 2 + 5 + 7, and day four 2+ 5 + 7 + 9. I think many of my students would have done it this way because they are much more comfortable doing adding than they are using multiplication. Of course, this would make skipping to day 17 a lot harder.

    • Jon

      April 6, 2022 at 6:28 am

      You’re right Terry! Many students would do it that way. If our learning goal however, was to help students develop a quadratic model (or algebraic representation) of this pattern then we’d want to help with generalizing. We often have students focus on the images in this example and ask “what would picture 10 look like”, “what would 17 look like?” “what would figure n look like?” “can you draw a diagram to show the shape?”

  • Amy Mathewson

    April 5, 2022 at 10:11 am

    I loved the multiple representations!

  • Laura Compean

    May 9, 2022 at 11:56 am

    So many teachers have always thought “their way” of explaining it is the best. Sadly, some teachers mark their students down when the student does it a different way. I think too many teachers haven’t been exposed to this multiple representations and are fearful the kids will stump them. I am no expert in seeing all these different ways but I’m excited to hear what the students say. They can literally teach me.

    • Kyle Pearce

      May 10, 2022 at 6:47 am

      Having an open stance to learn with your students is a fantastic approach to make this work in your classroom. Thanks for sharing!

  • Jared Sliger

    June 8, 2022 at 6:27 pm

    I presented at a conference last week and was having the group of teachers solve the marching band problem. It was really need to see how many different approaches they took to solve the problem. I know they work better than students, I think their level of curiosity has helped them get to where they are, but they were great and they tried lots of different drawings.

    It was great to be able to come back together and look at all the really interesting methods and find one that we really liked.

    On a side note, my daughter has the same shirt, I’ll see if I can find it.

  • Colegio Markham

    June 9, 2022 at 1:09 pm

    how lucky to have been taught this way! I always learnt and taught it in symbols and I can see that I even understand it much better this way. I can SEE the squares and the whole pattern. Before it was just a method. Total epiphany. I am looking forward to relearn it like visually and what better way to learn something than teaching it!

    • Kyle Pearce

      June 10, 2022 at 6:51 am

      Sadly we weren’t taught this way nor did we teach this way for many years. However, now that we know better, we can do better!

  • Christine Pomatto

    June 16, 2022 at 1:37 pm

    I love this example! I texted a picture of it to my grade-level team to see if they could solve it. I mentioned this in my last response, but this is really helping me see the value of algebra tiles. Having the visual (and being able to show it in multiple ways) helps students understand what all those numbers and symbols mean.

    I could see this problem as the type of thing we come back to every once in a while. At the beginning of the year, maybe they would be solving using a purely visual approach (drawing pictures or even using manipulatives). Later in the year, it could apply to area or algebraic expressions (or both).

  • Renee Holmquist

    June 20, 2022 at 3:52 pm

    All students have different learning styles, don’t push them to use what makes sense to YOU as a teacher. Allow them to see their peers thinking and reasoning so they can learn from each other. Also breaking up that heart pattern allowed for students to see different ways to represent one function for the amount of hearts.

  • Deanna Semyon

    June 23, 2022 at 11:41 am

    What was your big take away from this lesson?

    Use multiple representations and let the students work through their process to explain.  This will increase understanding and validate their thinking.

    How might you use the tool discussed?

    I can see myself using this daily.  Working with students with a learning disability I need to be flexible in my thinking and introducing their strengths in math through using their abilities.  Many of the students do get math, but from a untraditional point of entry.  It may take longer, but it will be worth it so that students reduce math fear/anxiety and build resilience for all math skills.

    What is something you are still wondering?

    I just need to build my confidence that I can look at things quickly and help students move through the process.  I wonder what an appropriate response is when I do not follow the student process, yet they get to the answer.
  • Heidi Cheng

    June 25, 2022 at 2:39 am

    This is a great reminder that again, encouraging visual representations helps students to see the math clearly instead of just numbers and words in a problem. Thank you!

  • Stefania Lambusta

    June 29, 2022 at 10:26 am

    I love the multiple representations and including visual and algebraic ways of thinking. I try to do this during lessons and have students share. This is where I see student differentiated thinking. Some students will stick to concrete examples and others are able to stretch their thinking. I find it so valuable for students to see what their classmates come up with. I want to work on preparing some additional representations to share with the class.

  • Anna Clark

    July 2, 2022 at 3:10 pm

    So I actually had to take two entire classes on patterns in college and each class we just worked on a different pattern. The HARDEST part for me in the beginning was to find more than one way to see the pattern! I always loved it when I found the pattern with a long rule, then I would simplify the rule by combining like terms, look at the simplest form (in the heart pattern case the rule would be n^2+2n-) then look back at the pattern to see if I could see where it was happening geometrically.

    All that to say, it trained me to ALWAYS look for another way to see things. And it’s something I hope to show my students as well! It’s not just a good math challenge to get them to back up and see something in a different way than theirs, it’s also a great life lesson.

  • Julie Gonzales

    July 6, 2022 at 5:48 pm

    My big take away from this particular problem was the visual was very powerful for me to get all of the different representations. I was taught to put the function in a table and find the difference between terms. I saw the first difference was 5, then 7, then 9, etc. When the primary difference is not the same, then you find the secondary difference between primary differences (5, 7, 9, etc) with all of those being the same (2), then you know the function is quadratic. Then you would build the rule by guess and check. Soooooo many steps in order to come up with the rule.

    After I saw the visual (after the hint to organize in the shape of a square), I could get to the “nth” term very quickly. Talk about mind blown here! This was so much more easy than the way I was taught to memorize all those years ago!

  • Marion Mulgrew

    July 13, 2022 at 11:38 am

    When I do an activity like this, I love seeing the different ways kids can work out a solution. Then having them share on the board or via some online program is great. I am a big fan of visuals and then being able to use symbols to describe them. It always amazes kids what they were able to do. I used Steve Wyborney’s Splat puzzles, and had students create their own. They were beautiful, but I didn’t do a good enough job the first time, connecting the pictures to the symbols. I was so surprised when students who did a great job on splat puzzles found 2 step equations confusing!

    For the multiple representations, do you ever get students who are resistant to finding another way once they have “the answer”? How do you handle that?

    • Jon

      July 15, 2022 at 6:51 am

      Early in the year definitely! A good line we use before we verify if anything is correct is to ask them “How do you know you’re right? How can you convince me in another way”. Overtime students will come to expect this.

      • Marion Mulgrew

        July 15, 2022 at 2:36 pm

        I will use that. I have a sign to “be a math lawyer” to defend your solutions, but I like this too. It’s friendly wording.

  • Nicolle Ristow

    July 15, 2022 at 6:38 pm

    As a teacher we always tell students there are multiply ways of solving a problem but then don’t give time to different representation. I definitely have to do a better job of this.

    One thing I am going to do is try to show function with different representations:

    1. Visual: Table and Graphs

    2. Function notation

    3. Simplify and solve

    All three of these were shown in the example in the video.

  • Heather Alden

    August 1, 2022 at 12:16 pm

    I think valuing all student thinking is important and honoring multiple representations does this. Instead of just trying to get students to think about it “the right way” or the most efficient way, it pushes all students to think more flexibly and celebrates all student work.

  • Kami Fevery

    August 3, 2022 at 2:20 pm

    Funny I never could understand how patterning expressions worked. As a teacher myself, I one day just sat down and figured it out (the relationship between the term # and the value), but it was not easy. Not sure how I got through High School Math but ya know. I only wish this was the way I was taught as I know it would have definitely been less of a struggle for me. I have tossed around this notion of using visuals to understand how to construct expressions in my own class. This lesson though really reminded me the multiple ways that you can analyze the visual pattern and break it down. Leaving it open for our class to identity the number of ways you can visually deconstruct the pattern will be a must for me!

  • Kristie D’Arcangelo

    August 8, 2022 at 4:47 pm

    Although I did eventually come to the realization that it was the number plus 1 squared, minus two, I saw the visual expanding in a way that I think my students would see it. I looked at the first day with the 2 and realized that a “border” if you will, was put along the top and side to “protect” the original two hearts. Then that new shape had the same thing happen to it; essentially the previous shape had a squared corner added to itself with one more in the length/width each time. So the 2 hearts had a 3×3 corner added on with a shared corner heart for the 3s on each side. The 3rd diagram had a 4×4 added on and when I noticed that I also came to the conclusion that it was the square of the length minus 2 and that the length was one more than the number of the day. I am truly looking forward to working on problems like this with my students as it will certainly open their eyes to some of the more advanced math but in a truly concrete visual way. For students who have fine motor struggles, I would probably provide some manipulatives to use rather than drawing so many tiny shapes.

    • Kyle Pearce

      August 9, 2022 at 6:34 am

      This is one of our favourite tasks to use with groups of various ages. Super low floor and high ceiling. Lots of math fun to be had here!

  • Craig Polzen

    November 7, 2022 at 10:30 am

    Thanks for sharing this Kyle… such a great task inspired by an everyday interaction.

    For me, the biggest takeaway was actually seeing the productive struggle in action with your family. I think this is the biggest challenge that we teachers face… how to ask questions without giving away too much… but giving enough to help them move forward.

  • Alison Peternell

    November 9, 2022 at 6:09 am

    I love growing patterns. I do this quite a bit with my 8th Grade Math and Algebra classes. I will say, the kids that have done something like this before, it is really hard to get them to not go straight to the Algebra representation and give away the final answer to the rest of the class. But, I tell them they did not follow the directions because I always start with the instruction to make a visual representation and then find how they think the pattern grows visually. I have made a slide deck that contains growing patterns so I can easily pop one up on the overhead projector to try them out periodically. I will add the picture of your daughter and her shirt to it and try it with my class.

    As far as representing things in multiple representations, this is good teaching. It helps deepen understanding and helps students who may struggle to see something one way, see it better another. It also adds to flexible thinking.

  • Victoria Murphy

    November 12, 2022 at 3:24 pm

    I show growing patterns a lot in my 7th grade class. We start with just talking about how is it growing and how you might find the number of stars or whatever the shape is in some small number and then a bigger number. But I did not think about breaking the shape apart in different ways to show equivalent expressions. I am looking forward to trying that with our next unit of expressions and equations. (And what a great way of giving an entry point for multiple levels of kids, we can show that the same pattern can be represented in a variety of ways and they are all equivalent!)

  • Noel McMillin

    November 21, 2022 at 5:46 pm

    I loved just how many different ways I could represent the pattern (with your encouragement). I can see spending more than one class period on this one problem. And I think the advantage of taking the time for multiple representations is the students who just didn’t get there the first time, get a second chance to feel successful.

    My first solution was similar to the parents’ solution. I multiplied 7 (day 2) by 14 (day 3) for 98 hearts. The problem with that pattern is there is no basis for day 1 and day 2, 2 hearts on day one and 7 hearts on day 2 are completely arbitrary. I can see how it is important to spend time with the problem before handing it off to students just so I have a handle on the different paths students can take.

  • Joseph Barnas

    December 2, 2022 at 1:57 pm

    The best part about these types of problems is that the multiple methods of representation are not obvious. We often talk in our classes about multiple representations and different avenues of getting an answer, but typically students only use one, maybe two ways to solve a problem. I wonder how many different representations would be present in my classroom.

  • Rebekah Schofield

    December 15, 2022 at 1:29 pm

    My biggest take away was that the visual representation is what made everything fall into place with me. I was able to figure out the pattern, but it wasn’t until I saw the graphics that I was able to write it Algebraically. I have always struggled with this type of problem, and now I have a tool to help. me.

    I have been using several different websites such as Esti-Mystery, and Mashup Math, and Open in the middle, as entry task lessons. The students really like them and so will try the problems and have discussions about the problems. Each day I make connections and fall ever more in love with math! What I have been learning here is definitely helping me to help my students.

    When I watch your videos and participate, then I am able to see the possibilities of the problems, but if I am just looking for problems myself, I struggle to see all the possibilities. I was taught to follow steps and to work algorithms, I am seeing what a disservice that was to me and I don’t want to pass it on to my students. But it is taking me so long to make the switch because I don’t have the experiences. Any suggestions?